Question: $f(x, y) = 3x + y^2 - y$ We have a change of variables: $\begin{aligned} x &= X_1(u, v) = 5u - 2v \\ \\ y &= X_2(u, v) = v - u \end{aligned}$ What is $f(x, y)$ under the change of variables? Choose 1 answer: Choose 1 answer: (Choice A) A $16u - 6v + 4v^2 - 8uv + 4u^2$ (Choice B) B $14u - 7v + v^2 - uv + u^2$ (Choice C) C $14u - 5v + v^2 - 2uv + u^2$ (Choice D) D $16u - 7v + v^2 - 2uv + u^2$
Solution: When applying a change of variables, we substitute the new definition for $x$ and $y$ into the original equation. The original equation: $f(x, y) = 3x + y^2 - y$ Let's substitute $X_1(u, v)$ for $x$ and $X_2(u, v)$ for $y$. $\begin{aligned} f(x, y) &= 3 \left( 5u - 2v \right) + \left( v - u \right)^2 - \left( v - u \right) \\ \\ &= 15u - 6v + v^2 - 2uv + u^2 - v + u \\ \\ &= 16u - 7v + v^2 - 2uv + u^2 \end{aligned}$ Therefore, under the change of variables, $f(x, y)$ becomes: $16u - 7v + v^2 - 2uv + u^2$